Cryptology research track for WS-CAND-003

Status: exploratory research only

Date opened: 2026-08-14

Scope: possible cryptologic relevance of Type A/B witness depth, trap geometry, and congruence shadowing

This note records a cryptology-facing research program arising from WS-CAND-003. It does not claim a break of RSA, discrete logarithm systems, elliptic-curve cryptography, post-quantum cryptography, or any deployed cryptosystem.

The motivating observation is that the current work has produced a deterministic arithmetic fingerprint of primes and an exact redundancy structure among modular constraints:

with

plus a direct-shadow relation describing when one modular trap layer is implied by an earlier layer.

Cryptology routinely depends on prime generation, modular arithmetic, structured-prime selection, sieving, and the absence of unintended arithmetic bias. This makes the new objects worth testing as cryptographic diagnostics and sieve structure, even though no cryptographic impact has yet been demonstrated.

1. The strongest plausible connection: prime-source fingerprinting

C_AB(p) and the fuller hit/shadow profile of a prime can be treated as arithmetic features.

Define a finite fingerprint through depth K by

The first cryptologic question is whether the distribution of these features differs detectably between prime populations generated under different structural constraints.

Candidate populations include:

• unconstrained random probable primes of a fixed bit size;
• Blum primes p == 3 mod 4;
• safe primes p=2r+1 with r prime;
• primes restricted to selected residue classes;
• primes produced by standards-oriented generation procedures, when reproducible test implementations are available;
• deliberately structured toy prime families used as controls.

A measurable distinction would not itself imply weakness. It would establish that the Type A/B depth/shadow system is sensitive to prime-generation structure and could therefore serve as a prime-distribution audit statistic.

2. Prime-generation auditing and backdoor detection

Modern public-key systems depend on suitable prime or prime-field parameters. Standards such as NIST FIPS 186-5 specify RSA and signature-related parameter generation requirements, and NIST validation material explicitly treats properties of generated RSA primes as testable key-generation outputs.

The research hypothesis is:

If a prime generator introduces hidden modular structure, a Type A/B depth or shadow fingerprint may detect distributional deviations even when ordinary low-modulus checks do not.

This is a detection hypothesis, not an assertion that any standardized generator is biased or vulnerable.

A useful negative control is equally important: if C_AB fingerprints are statistically indistinguishable after conditioning on the generator's explicit congruence requirements, that limits the cryptologic relevance of the invariant in this direction.

3. Structured-prime diagnostics

Cryptanalytic history contains many examples in which special arithmetic structure in parameters matters. Number-field-sieve variants exploit special-form finite-field primes, and lattice/Coppersmith techniques can exploit particular modular polynomial structure in vulnerable constructions.

The current Type A/B framework should therefore be tested as a structure detector, not assumed to be an attack.

Questions:

1. Do primes with unusually large or small C_AB exhibit correlations with known structured-prime families?
2. Does the ancestry/shadow profile detect arithmetic regularity not captured by a handful of small congruence tests?
3. Are record-depth primes unusually generic, unusually structured, or neither under standard prime statistics?
4. Can a shadow-compressed fingerprint identify deliberately planted toy prime-generation biases?

No positive answer should be promoted to a cryptographic weakness without a concrete reduction, distinguisher, key-recovery improvement, or parameter-generation failure.

4. Modular-sieve compression

The direct-shadow relation has an immediate algorithmic interpretation:

if one modular condition is implied by earlier modular conditions, it can be removed from a sieve without changing the accepted set.

The Type A/B system is not the number field sieve, quadratic sieve, or a cryptographic lattice attack. However, the mathematical pattern is close enough to motivate a generic question:

The research should separate two levels:

• domain-specific result: exact compression of the Type A/B congruence sieve;
• transfer result: a generalized shadow criterion that improves a cryptologically relevant sieve or modular search problem.

Only the second would establish a direct cryptologic algorithmic impact.

5. RSA: what can and cannot currently be said

For an RSA modulus

the factors p and q are secret, while N is public. A factor-specific value such as C_AB(p) is therefore not directly observable from the public key.

The current research does not provide a factorization algorithm.

There is nevertheless a concrete research question. For every k, public knowledge of

couples the two unknown factor residues. The trap sets T_k, together with cross-layer ancestry and shadow constraints, define finite residue systems for the possible factors.

The experimental question is whether combining these public product constraints across many irredundant layers reduces the candidate factor-residue space faster than an appropriate random baseline.

This must first be tested on toy RSA moduli where the true factors are retained only for scoring. A useful outcome may be either positive or negative:

• positive: a measurable residue-space reduction beyond trivial congruence information, motivating deeper cryptanalysis;
• negative: evidence that the factor-specific invariant does not leak usefully through the product modulus.

Until such an experiment succeeds, describing the work as an RSA attack would be incorrect.

6. Public-modulus fingerprint

For a modulus N and layer k, define the public product residue

One simple diagnostic is the product-trap set

Then

is computable without factoring N.

This does not prove that the actual secret factors are both in T_k; it only tests compatibility with that possibility. The purpose of G_K is empirical: determine whether different prime-generation families induce distinguishable public-modulus distributions after controlling for obvious residue constraints.

7. Immediate experiments

The cryptology track begins with deliberately modest, falsifiable experiments:

1. Generate equal-size samples of random primes, Blum primes, safe primes, hard-class primes, and deliberately biased toy primes.
2. Measure C_AB, hit-vector, direct-shadow, and ancestry statistics for every prime.
3. Compare empirical distributions using total-variation distance and simple held-out classification, with explicit controls for known congruence restrictions.
4. Form toy RSA moduli from each population and compute public product-trap fingerprints G_K(N).
5. Measure whether the public fingerprints identify the source population beyond the information already implied by low-modulus constraints.
6. Test whether removing directly shadowed layers preserves all measured information while reducing work.
7. If a nontrivial signal survives, escalate to larger bit sizes and stronger statistical tests.

The initial experiments are diagnostic. They must not be described as key recovery or cryptographic breaks.

8. CENTL's role

CENTL should remain the exact-verification layer rather than the statistical oracle.

For each cryptology experiment it should certify, where expressible in the current language:

• exact algebraic identities used to derive structured prime families;
• modulus-ancestry polynomial identities;
• exact rational summaries and derived formulas;
• generated proof contracts for any claimed algebraic reduction.

Python or another dedicated finite-search engine may enumerate primes and residue classes, but the resulting algebraic claims should be replayed through CENTL and accompanied by deterministic artifacts and hashes, following the existing WS-CAND-003 methodology.

9. Claim discipline

Permitted current wording:

The Type A/B witness-depth and shadow framework has plausible cryptologic relevance as a new arithmetic fingerprint of primes and as a method for identifying redundant modular constraints. FCF is testing whether that structure yields useful prime-generation distinguishers, parameter-audit diagnostics, or modular-sieve compression.

Not currently justified:

• "FCF broke RSA."
• "The Erdős-Straus work weakens modern encryption."
• "C_AB leaks RSA factors."
• "The shadow graph improves the number field sieve."
• "Cryptographic standards generate weak primes."

Those require separate evidence.

10. Primary references for the cryptology bridge

• NIST, FIPS 186-5: Digital Signature Standard, 2023, https://doi.org/10.6028/NIST.FIPS.186-5
• NIST ACVP RSA key-generation specification, including generated-prime properties and test modes, https://pages.nist.gov/ACVP/draft-celi-acvp-rsa.html
• R. Barbulescu, P. Gaudry, T. Kleinjung, The Tower Number Field Sieve, IACR ePrint 2015/505. This is cited only as established evidence that special arithmetic form of cryptographic parameters can affect sieve complexity, not as evidence that C_AB has such an effect.
• H. Davis, M. Green, N. Heninger, K. Ryan, A. Suhl, On the Possibility of a Backdoor in the Micali-Schnorr Generator, IACR ePrint 2023/440. This is cited as an example of cryptanalysis exploiting structured parameters with lattice techniques, not as an analogue already established for the present framework.

11. Research target

The strongest near-term cryptology result would be one of the following:

1. a reproducible distinguisher showing that F_K detects a nontrivial prime-generation bias after conditioning on all obvious congruence information;
2. a public-modulus statistic derived from the shadow-compressed system that distinguishes structured toy RSA populations without access to the factors;
3. a general shadow-compression theorem that transfers to a recognized cryptologic modular sieve.

Until one of these is obtained, the cryptology connection remains a motivated research program rather than a cryptographic impact claim.